
@ARTICLE{Shakan2019,
       author = {George Shakan},
        title = "{On higher energy decompositions and the sum─product phenomenon}",
      journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
     keywords = {Mathematics - Number Theory, Mathematics - Combinatorics, 11B13 11B30 11B75},
         year = "2019",
        month = "Nov",
       volume = {167},
       number = {3},
        pages = {599-617},
          doi = {10.1017/S0305004118000506},
archivePrefix = {arXiv},
       eprint = {1803.04637},
 primaryClass = {math.NT},
       adsurl = {https://ui.adsabs.harvard.edu/abs/2019MPCPS.167..599S},
      adsnote = {Provided by the SAO/NASA Astrophysics Data System}
}

@Inbook{Erdos1983,
       author = "Erd{\H{o}}s, P.  and Szemer{\'e}di, E.",
       editor = "Erd{\H{o}}s, Paul and Alp{\'a}r, L{\'a}szl{\'o} and Hal{\'a}sz, G{\'a}bor and S{\'a}rk{\"o}zy, Andr{\'a}s",
        title = "On sums and products of integers",
    bookTitle = "Studies in Pure Mathematics: To the Memory of Paul Tur{\'a}n",
         year = "1983",
    publisher = "Birkh{\"a}user Basel",
      address = "Basel",
        pages = "213--218",
     abstract = "Let 1 ≦ a1 <... <anbe a sequence of integers Consider the integers of the form1{\$}{\$}a{\_}i + a{\_}j ,{\backslash},a{\_}i a{\_}j ,{\backslash},1 {\backslash}leqq i {\backslash}leqq j {\backslash}leqq n.{\$}{\$}",
isbn="978-3-0348-5438-2",
          doi = "10.1007/978-3-0348-5438-2_19",
          url = "https://users.renyi.hu/~p_erdos/1983-18.pdf"
}


